The total \(\chi ^2\)  is defined as \(\chi ^2=\chi _{CMB}^2+\chi _{SN}^2+\chi _{BAO}^2+\chi _{H}^2+(\Omega _{r0}-9.13\times 10^{-5})^2/(0.25\times 10^{-5})^2\) . The priors used for the MCMC analysis are \(\Omega _{m0}\in \left[0.2,0.4\right]\) , \(\Omega _{r0}\in \left[10^{-5},10^{-4}\right]\) , \(\Omega _{b0}h_0\in \left[0.001,0.06\right]\)  and \(h_{0}\in \left[0.55,0.80\right]\) . Figure REF  shows the \(68.3\%\) , \(95.4\%\)  and \(99.7\%\)  confidence contours for the coupled model and the one-dimensional marginalized likelihoods got with a sample of 500000 MCMC points (the black points are the mean MCMC values).At \(68.3\%\) , we have \(\Omega _{m0}=0.298\pm 0.006\) , \(10^3\Omega _{r0}=0.091\pm 0.002\) , \(\Omega _{b0}h_0^2=0.0224\pm 0.0001\) , \(\alpha =5.42\pm 0.13\)  and \(H_0=67.74\pm 1.17\) , the best \(\chi ^2\)  being \(1068.07\) . The best \(\Lambda CDM\)  and coupled models \(\chi ^2\)  with their components are shown in table REF . Using the Akaike information criterion (AIC)[1]} as done in [2]} and [3]}, we can compare the two models. Both of them being constrained by the same 1098 data and the coupled model having one more free parameter than the \(\Lambda CDM\)  one, we find that
\(AIC_{coupled}-AIC_{\Lambda CDM}\simeq 2.7\) 
